Intersection with the vertical isocline in the Liénard plane (Q943764)

The authors study the problem of whether all trajectories of the Liénard system \[ \dot{x}=y-F(x),\quad \dot{y}=-g(x) \] intersect the vertical isocline \(y=F(x)\). This property is a crucial step for the proof of the existence of oscillatory and periodic solutions and for the global asymptotic stability of the zero solution. As a consequence of the results presented in the paper it is shown that the conjecture given by \textit{T. Hara} and \textit{J. Sugie} in [NoDEA, Nonlinear Differ. Equ. Appl. 2, No.~4, 527--551 (1995; Zbl 0840.34033)] is true.